19] 
D> yzi=jz1—kyz*, 
D2 yz? =- 2! yz, 
Ds yzt=—2!j23+3! hyzt, 
Dt yzt=4! yz, 
Di yzt=4! jz5-5! kyz, 
* ee e @ =e Yeitie  aete 
Hence, we find 
3 
mw yz =ye + xv (jer — kyz*) — xyz? — a (2! j23-3! kyz*) 
5 
+ wtyz?+ - (4! jz - 51 kyz*) — a®yz" — &e. 
Now let us compare this with the development of 
(y+jz) (2+kx). 
Expanding by the binomial theorem, and preserving the 
powers of j and &, when both appear in the same coefficient, 
we have 
y+ je) (2+ hay =yo? + x (jet — hye) — a (ghz? + yz 
J J J y 
+ 28 (jk228 + hyz*) — a4 (jh8z4 — yz) 
B) 
+ a5 (jhte’ — hyz*)— 2° (jheet+ yz") . . & 
The discrepancies between the developments (A) and (B) are 
numerous, but all of them are of the same kind. In the first 
place the terms 2%z, xtz4, a°z*, &e., do not appear in (A). 
In (B) they have the coefficients jh, jh®, jh°, &e. But the mean 
values of such products of j and & are equal to zero. 
‘¢ Again, the mean value of the product of one j and 2» 
1 
2v+1 
ments of a°z*, wz, &c., differ just in this: that in (A) they 
are mean products, in (B) ordinary products of js and és. 
k 8 18, (-1). Hence the coefficients in the two develop- 
Thus it appears, as we anticipated, that if we substitute mean 
products of js and ’s for ordinary products throughout the 
